Beyond the personal loss, Harry Markowitz leaves an indelible mark on finances that we want to celebrate with these lines (another sample, here). And how better to celebrate it than by sharing his most fundamental contribution: the theory of efficient portfolio selection, and the ideas that underlie said theory. Some of these ideas are not at all obvious, but they seem so largely because they are already integrated into our daily lives.
Markowitz makes his mark on finance as a catalyst for the fundamental change that occurred in the 1950s. Before 1950, finance was the domain of powerful tycoons, where investment decisions were made based on what a company could earn and Risk was a nebulous concept that only a privileged few could master. We talk about “the finances of private clubs.” With Markowitz begins the transition to academic finance, to its development as a science, with its logic embodied in mathematical formulas and relationships and, therefore, open to anyone who has basic mathematical and statistical foundations.
Nowadays, we take it for granted that with a little effort you can invest on your own without too much trouble, and we don’t give it any importance. But pre-Markowitz this was not the case, and before H. Markowitz’s footprint fades completely, like that of those who invented writing, we take advantage of this moment to recognize his important catalytic role. Let’s start with the historical context from which the change arises. Our readers, especially the youngest ones, must be transported to a now-distant world: the decade that begins in 1950. We begin on April 21, 1951. At MIT, the first computer that works in real-time is built, called Whirlwind. (whirlwind, in Spanish). A computer that weighs more than 900 kilos, that crashes every 20 minutes on average and that has a memory capacity of no more and no less than 2k (not Gigs, not megabytes, two magnificent Kilobytes – a single Emoji takes up 12 Kilobytes!). Harry Markowitz arrived in this decade, enrolling at the University of Chicago, where he completed his undergraduate and master’s studies in mathematics applied to economics, and read his doctoral thesis in 1954. The contributions of his doctoral thesis were so novel that his comment became famous. Milton Friedman suggested that “that” was not a thesis in economics and, therefore, the doctorate title could not be recognized.
And how could it be that what “was not economics” received the Nobel Prize in economics in 1990? The basic topic of the thesis was investment analysis. Everyone, even the financial magnates in their private clubs, already knew that putting all their eggs in one basket was not a very good idea. The investments had to be distributed, at least a little. How much? Well, we don’t know, a little bit here, a little bit there. Here comes Markowitz. Guided by the mathematical concept of efficiency transmitted to him by his professor Tjalling Koopmans (Nobel Prize in economics in 1975), he applied mathematics to define the investment problem, defining an objective criterion to compare when one portfolio is “better” than another. , and thereby establish the best investment strategy. This sounds like a lot of math, But as we had mentioned before, the basic concepts are within the reach of any investor. And we are going to try to convey these concepts even without mathematics.
First of all, the simile of eggs in the basket is not good to describe the choice of companies with whose shares to create an investment portfolio. It would be more appropriate to think that it is about choosing which chickens are suitable to maintain a good poultry farm, because the secret is not in the companies themselves (the eggs or chickens), but in how the share prices of a company are related to the others (whether one type of chicken gets along with the others or not).
That said, let’s move on to define the investment problem in more detail. An investment (in the stock market) involves using your money today to buy shares of one or more companies and, in the future, selling them again. This allows you to transfer the resources you can save today to the future when you will need to use those resources. But, between the purchase today and the future sale, the price of your investments can rise or fall, and that depends on all the things that can happen between now and then. Mathematically, the future price of the stock can be described as a random variable, that is, a mathematical object that meets certain statistical rules, such as having an expected value, and a variance. Markowitz established the following decomposition of investment: on the one hand, you have the expected return, the difference between the initial money to buy shares and the expected value of the selling price of the shares that you will get in the future. On the other hand, you have the difference between the expected sales value and the price you will actually receive. This deviation can be good (above the expected value) and bad (below the expected value), although on average the deviation is zero (by definition of the expected value). But this difference (between the expected sales value and the actual one) has an important property called variance. It is not the same as whether the deviations are +1 or -1, or whether they are +10 or -10. Both combinations of deviations from the mean are at zero mean, but in the second case, the deviations are much larger. That is described by the variance. The second combination (+10 or -10) has more variance than (+1 or -1).
For Markowitz, these two components, the expected return and the variance, are enough to describe what is important about investment and establish criteria for comparing portfolios. If the expected return is higher, that’s clearly better—your future self will have (on average) more to spend. But if the deviations are larger (+10, -10), it is worse if they are smaller (-1 or +1). So how do I compare two portfolios? Easy. If both have the same expected profitability, the one with less variance is better. But if we change the expected return, normally the portfolio with the lowest variance will be different. What interests us is the set of the best portfolios (the so-called efficient frontier). This set consists of making a list of all the expected return pairs and the portfolio with the lowest variance associated with that expected return, and choosing for each level of variance the return-portfolio pair with the highest expected return. It seems a little confusing, but that’s because in the list of pairs there are some pairs with the same variance and lower expected profitability, and we want to be very careful.
A careful reading shows that, along the way, we have changed the investment object. We start in private clubs choosing companies on an individual basis. Now we are talking about stocks in a portfolio. From Markowitz, the important thing is not whether a company is good or not (or at least it is not the most important thing). The important thing is what the company’s shares contribute to the investment portfolio. In terms of expected profitability, this is simple. A return above the average return of the portfolio increases the return of the portfolio. And vice versa if it is lower. It has a simple, linear effect. But the risk, the variance, no. When you add a new stock to a portfolio, its effect on the risk (variance) of the portfolio does not depend on the variance of the stock. Let us repeat this because it is very surprising: if you add a stock with a lot of variance it may or may not raise the variance. It may go down. The effect of the risk of a stock on the risk of the portfolio depends not on the variance of the stock but on its covariance with the rest of the stocks that are already present in the portfolio. Broadly speaking, it is a statistical concept that describes the relationship between the deviations from the mean of two variables. If we talk about the returns of two stocks, positive covariance indicates that when the return of one is above the mean, the return of the other will also be (statistically) above the mean. And vice versa, when the profitability of one is below average, so is that of the other. The covariance can also be negative—when one stock goes up relative to the mean, the other goes down—and the covariance can be higher or lower depending on how strong this interrelationship between returns is. In terms of a portfolio, the good thing is that they compensate: when the profitability of one stock is above its average, another is below, and thus they compensate, the profitability of the portfolio is closer to its average, and the Portfolio risk decreases. Returning to the chicken coop simile: the important thing is not the chicken itself, but how that chicken gets along with the ones already in the chicken coop. The smaller the covariance, and even better if it is negative, the better the chickens get along with each other. This was demonstrated by Markowitz. The other goes down—and the covariance can be higher or lower depending on how strong this interrelationship between the returns is. In terms of a portfolio, the good thing is that they compensate: when the profitability of one stock is above its average, another is below, and thus they compensate, the profitability of the portfolio is closer to its average, and the Portfolio risk decreases. Returning to the chicken coop simile: the important thing is not the chicken itself, but how that chicken gets along with the ones already in the chicken coop. The smaller the covariance, and even better if it is negative, the better the chickens get along with each other. This was demonstrated by Markowitz. the other goes down—and the covariance can be higher or lower depending on how strong this interrelationship between the returns is. In terms of a portfolio, the good thing is that they compensate: when the profitability of one stock is above its average, another is below, and thus they compensate, the profitability of the portfolio is closer to its average, and the Portfolio risk decreases. Returning to the chicken coop simile: the important thing is not the chicken itself, but how that chicken gets along with the ones already in the chicken coop. The smaller the covariance, and even better if it is negative, the better the chickens get along with each other. This was demonstrated by Markowitz. The portfolio’s return is closer to its average, and the portfolio’s risk decreases. Returning to the chicken coop simile: the important thing is not the chicken itself, but how that chicken gets along with the ones already in the chicken coop. The smaller the covariance, and even better if it is negative, the better the chickens get along with each other. This was demonstrated by Markowitz. The portfolio’s return is closer to its average, and the portfolio’s risk decreases. Returning to the chicken coop simile: the important thing is not the chicken itself, but how that chicken gets along with the ones already in the chicken coop. The smaller the covariance, and even better if it is negative, the better the chickens get along with each other. This was demonstrated by Markowitz.
These are Markowitz’s investing basics. Not even an equation! But it didn’t stop there. The next problem was how to obtain the basic data (expected returns, variances, and covariances), and once obtained, how to calculate the efficient portfolio. And remember, there is no single efficient portfolio—you’ve seen the contortionism we had to do to define the best portfolios, the efficient frontier. For each level of variance (risk), the one with the highest expected return will be an efficient portfolio. But like the title of our blog, nothing is free. If you want more expected returns, the variance of the efficient portfolio increases. Therefore, it is not necessary to calculate an efficient portfolio, but rather the entire efficient frontier. And so within this border,
Let’s remember that the thesis is from 1954. Do you remember Torbellino, the 2k supercomputer? With pencil and paper, Markowitz’s solution might be elegant, but having to identify so many optimal portfolios was not useful. And that’s where Markowitz continued. Not content with a contribution that would earn him the Nobel Prize, he set to work (at the Rand Corporation) on how to use new computing technology to apply his solutions. And from there came two great additional contributions: on the one hand, the SIMSCRIPT programming language, and on the other he obtained several fundamental results on sparse matrices ( sparse matrices).). SIMSCRIPT never became as impactful as the great Grace Murray Hopper’s COBOL language in 1959, but it is still used today in some flying radars. And sparse matrices are a fundamental tool to obtain efficient portfolios since the problem requires working with huge matrices and many zeros. For these works he received the John von Neumann Award from the American Society for Operations Research in 1989.
Today, the trace of Harry Markowitz’s academic contributions can be found in finance and beyond, hidden among lines of code increasingly hidden behind layers and layers of programming. The computers of the 21st century have democratized computing and with them the finances that this intellectual giant, Harry Markowitz, anticipated.